The low-energy physics of (quasi)degenerate one-dimensional systems is typically understood as the particle-like dynamics of kinks between stable, ordered structures. Such dynamics, we show, becomes highly non-trivial when the ground states are topologically constrained: a dynamics of the domains rather than on the domains which the kinks separate. Motivated by recently reported observations of charged polymers physisorbed on nanotubes, we study kinks between helical structures of a string wrapping around a cylinder. While their motion cannot be disentangled from domain dynamics, and energy and momentum is not concentrated in the solitons, the dynamics of the domains can be folded back into a one-particle picture.PACS numbers: 03.65. Vf, 64.70.Nd, The relationship between topological and physical properties [1][2][3] has received much recent attention. It is relevant to elasticity [3,4], non-linear physics [5,6], soft and hard condensed matter [2,3,7], and quantum computing [8,9]. Topological invariants associated to physical objects often dictate interaction: for instance punctures in a plane (defects, dislocations, vortices) define a topological invariant (the winding angle) and thus a logarithmic field which non surprisingly also to mediates their mutual interaction [4]. Similarly, topologically distinct states support infinitely continuum transitions [10,11].We have previously investigated [11] the statistical mechanics (and connections with conformal invariance in quantum mechanics) of topological transitions among winding states representing winding/unwinding polymers. Here we study the Newton dynamics of a selfinteracting string (polymer) winding around a cylinder (nanotube). If strings are stable in different, and non necessarily degenerate, helical structures, they exhibit topological solitons whose dynamics, however, is not "contained" in the kink but involve the entire system. This is a feature of the topology of helical solitons found also in systems of essentially different physics: in "dynamical phyllotaxis" [12,13] repulsive particles in cylindrical geometries mimic botanical patterns of leaves on stems, spines on cactuses, petals on a flower [15] by selforganizing in helical lattices described by Fibonacci numbers [12,14], also separated by kinks; or in colloidal crystals on cylinders and rod-shaped bacterial cell walls [16].While our analysis elucidates an interesting case of topology-dictated dynamics connected to the simplest topological invariant-the winding number-it is not without practical implications. Polymer-nanotube hybrids, ssDNA-carbon nanotubes in particular [17][18][19][20], have been the subject of much recent experimental and numerical research [17][18][19][20][21][22][23][24][25][26][27] as promising candidates for nanotechnological applications in bio-molecular and chemical sensing, drug delivery [17,28] and dispersion/patterning of carbon nanotubes [19][20][21]. Indeed, ss-DNA forms tight helices on carbon nanotubes after sonication of the hybrids, although the role of base de...
